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September  2021, 10(3): 545-573. doi: 10.3934/eect.2020080

Null controllability for singular cascade systems of $ n $-coupled degenerate parabolic equations by one control force

Brahim Allal 1, , Abdelkarim Hajjaj 1, , Lahcen Maniar 2,, and  Jawad Salhi 1,

1. 

Faculté des Sciences et Techniques, Université Hassan 1er, Laboratoire MISI, B.P. 577, Settat 26000, Morocco
2. 

Département de Mathématiques, Faculté des Sciences Semlalia, LMDP, UMMISCO (IRD-UPMC), Université Cadi Ayyad, Marrakech 40000, B.P. 2390, Morocco

* Corresponding author: Lahcen Maniar

Received  June 2019 Revised  March 2020 Published  September 2021 Early access  July 2020

In this paper, we consider a class of cascade systems of $ n $-coupled degenerate parabolic equations with singular lower order terms. We assume that both degeneracy and singularity occur in the interior of the space domain and we focus on null controllability problem. To this aim, we prove first Carleman estimates for the associated adjoint problem, then, we infer from it an indirect observability inequality. As a consequence, we deduce null controllability result when a unique distributed control is exerted on the system.

Keywords: Degenerate parabolic equations, singular cascade systems, Carleman estimates, observability inequality, null controllability.
Mathematics Subject Classification: Primary: 35K65, 35K67, 93B05; Secondary: 93C20, 47D06.
Citation: Brahim Allal, Abdelkarim Hajjaj, Lahcen Maniar, Jawad Salhi. Null controllability for singular cascade systems of $ n $-coupled degenerate parabolic equations by one control force. Evolution Equations & Control Theory, 2021, 10 (3) : 545-573. doi: 10.3934/eect.2020080
References:
[1]

F. Alabau-Boussouira, A hierarchic multi-level energy method for the control of bidiagonal and mixed $n$-coupled cascade systems of PDE's by a reduced number of controls, Adv. Differential Equations, 8 (2013), 1005-1072.   Google Scholar

[2]

K. Atifi, I. Boutaayamou, H. O. Sidi and J. Salhi, An inverse source problem for singular parabolic equations with interior degeneracy, Abstract and Applied Analysis, 2018 (2018), 1-16.  doi: 10.1155/2018/2067304.  Google Scholar

[3]

F. Ammar Khodja, A. Benabdallah, M. Gonzalez-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[4]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy Problems, Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.  Google Scholar

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U. Biccari, V. Hernandez-Santamaria and J. Vancostenoble, Existence and cost of boundary controls for a degenerate/singular parabolic equations, preprint, arXiv: 2001.11403. Google Scholar

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I. Boutaayamou, G. Fragnelli and L. Maniar, Lipschitz stability for linear parabolic systems with interior degeneracy, Electron. J. Differential Equations, 2014 (2014), 1-26.   Google Scholar

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I. Boutaayamou and J. Salhi, Null controllability for linear parabolic cascade systems with interior degeneracy, Electron. J. Differential Equations, 2016 (2016), 1-22.   Google Scholar

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P. Cannarsa, G. Floridia, F. Golgeleteyen and M. Yamamoto, Inverse coefficient problems for a transport equation by local Carleman estimate, Inverse Problems, 35 (2019), 22 pp. doi: 10.1088/1361-6420/ab1c69.  Google Scholar

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P. Cannarsa, R. Ferretti and P. Martinez, Null controllability for parabolic operators with interior degeneracy and one-sided control, SIAM J. Control Optim., 57 (2019), 900-924.  doi: 10.1137/18M1198442.  Google Scholar

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P. Cannarsa and L. De Teresa, Controllability of $1$-D coupled degenerate parabolic equations, addendum and corrigendum, Electron. J. Differential Equations, 2009 (2009), 1-24.   Google Scholar

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M. Duprez and P. Lissy, Indirect controllability of some linear parabolic systems of $m$ equations with $m-1$ controls involving coupling terms of zero or first order, J. Math. Pures Appl., 9 (2016), 905-934.  doi: 10.1016/j.matpur.2016.03.016.  Google Scholar

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M. Fadili and L. Maniar, Null controllability of $n$-coupled degenerate parabolic systems with $m$-controls, J. Evol. Equ., 17 (2017), 1311-1340.  doi: 10.1007/s00028-017-0385-3.  Google Scholar

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E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations, J. Funct. Anal., 7 (2010), 1720-1758.  doi: 10.1016/j.jfa.2010.06.003.  Google Scholar

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G. Floridia, C. Nitsch and C. Trombetti, Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states, ESAIM Control Optim. Calc. Var., 26 (2020), 34 pp. doi: 10.1051/cocv/2019066.  Google Scholar

[20]

M. Fotouhi and L. Salimi, Controllability results for a class of one dimensional degenerate/singular parabolic equations, Commun. Pure Appl. Anal., 12 (2013), 1415-1430.  doi: 10.3934/cpaa.2013.12.1415.  Google Scholar

[21]

M. Fotouhi and L. Salimi, Null controllability of degenerate/singular parabolic equations, J. Dyn. Control Syst., 18 (2012), 573-602.  doi: 10.1007/s10883-012-9160-5.  Google Scholar

[22]

G. Fragnelli, Interior degenerate/singular parabolic equations in nondivergence form: Well-posedness and Carleman estimates, J. Differential Equations, 260 (2016), 1314-1371.  doi: 10.1016/j.jde.2015.09.019.  Google Scholar

[23]

G. Fragnelli, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Generators with interior degeneracy on spaces of $L^2$ type, Electron. J. Differential Equations, 2012 (2012), 1-30.   Google Scholar

[24]

G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients, Adv. Nonlinear Anal., 6 (2017), 61-84.  doi: 10.1515/anona-2015-0163.  Google Scholar

[25]

G. Fragnelli and D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, Mem. Amer. Math. Soc., 242 (2016), 84 pp. doi: 10.1090/memo/1146.  Google Scholar

[26]

G. Fragnelli and D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Adv. Nonlinear Anal., 2 (2013), 339-378.  doi: 10.1515/anona-2013-0015.  Google Scholar

[27]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of evolution equations, Lect. Notes Ser., 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar

[28]

M. González-Burgos and L. De Teresa, Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force, Portugal. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.  Google Scholar

[29]

A. Hajjaj, L. Maniar and J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electron. J. Differential Equations, 2016 (2016), 1-25.   Google Scholar

[30]

A.Y. Khapalov, Global non-negative controllability of the semilinear parabolic equation governed by bilinear control, ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 269-283.  doi: 10.1051/cocv:2002011.  Google Scholar

[31]

J. Le Rousseau and G. Lebeau, On carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.  Google Scholar

[32]

J. Salhi, Null controllability for a singular coupled system of degenerate parabolic equations in nondivergence form, Electron. J. Qual. Theory Differ. Equ., 13 (2018), 1-28.  doi: 10.14232/ejqtde.2018.1.31.  Google Scholar

[33]

J. Vancostenoble, Global non-negative approximate controllability of parabolic equations with singular potentials, in Trends in Control Theory and Partial Differential Equations, , Springer INdAM Series, 32, Springer, Cham, 2019,255–276.  Google Scholar

[34]

J. Vancostenoble, Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 761-790.  doi: 10.3934/dcdss.2011.4.761.  Google Scholar

show all references

References:
[1]

F. Alabau-Boussouira, A hierarchic multi-level energy method for the control of bidiagonal and mixed $n$-coupled cascade systems of PDE's by a reduced number of controls, Adv. Differential Equations, 8 (2013), 1005-1072.   Google Scholar

[2]

K. Atifi, I. Boutaayamou, H. O. Sidi and J. Salhi, An inverse source problem for singular parabolic equations with interior degeneracy, Abstract and Applied Analysis, 2018 (2018), 1-16.  doi: 10.1155/2018/2067304.  Google Scholar

[3]

F. Ammar Khodja, A. Benabdallah, M. Gonzalez-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[4]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy Problems, Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.  Google Scholar

[5]

J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constant formula, Proc. Amer. Math. Soc., 63 (1977), 370-373.  doi: 10.2307/2041821.  Google Scholar

[6]

U. Biccari, V. Hernandez-Santamaria and J. Vancostenoble, Existence and cost of boundary controls for a degenerate/singular parabolic equations, preprint, arXiv: 2001.11403. Google Scholar

[7]

I. Boutaayamou, G. Fragnelli and L. Maniar, Lipschitz stability for linear parabolic systems with interior degeneracy, Electron. J. Differential Equations, 2014 (2014), 1-26.   Google Scholar

[8]

I. Boutaayamou and J. Salhi, Null controllability for linear parabolic cascade systems with interior degeneracy, Electron. J. Differential Equations, 2016 (2016), 1-22.   Google Scholar

[9]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.  Google Scholar

[10]

P. Cannarsa, G. Floridia, F. Golgeleteyen and M. Yamamoto, Inverse coefficient problems for a transport equation by local Carleman estimate, Inverse Problems, 35 (2019), 22 pp. doi: 10.1088/1361-6420/ab1c69.  Google Scholar

[11]

P. Cannarsa, R. Ferretti and P. Martinez, Null controllability for parabolic operators with interior degeneracy and one-sided control, SIAM J. Control Optim., 57 (2019), 900-924.  doi: 10.1137/18M1198442.  Google Scholar

[12]

P. Cannarsa and L. De Teresa, Controllability of $1$-D coupled degenerate parabolic equations, addendum and corrigendum, Electron. J. Differential Equations, 2009 (2009), 1-24.   Google Scholar

[13] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon Press, Oxford, 1998.   Google Scholar
[14]

John B. Conway, A Course in Functional Analysis, 2$^nd$ edition, Springer-Verlag, New York, 1990.  Google Scholar

[15]

M. Duprez and P. Lissy, Indirect controllability of some linear parabolic systems of $m$ equations with $m-1$ controls involving coupling terms of zero or first order, J. Math. Pures Appl., 9 (2016), 905-934.  doi: 10.1016/j.matpur.2016.03.016.  Google Scholar

[16]

M. Fadili and L. Maniar, Null controllability of $n$-coupled degenerate parabolic systems with $m$-controls, J. Evol. Equ., 17 (2017), 1311-1340.  doi: 10.1007/s00028-017-0385-3.  Google Scholar

[17]

E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations, J. Funct. Anal., 7 (2010), 1720-1758.  doi: 10.1016/j.jfa.2010.06.003.  Google Scholar

[18]

J. Carmelo Flores and L. de Teresa, Null controllability of one dimensional degenerate parabolic equations with first order terms, Discrete & Continuous Dynamical Systems - B, 22 (2017). doi: 10.3934/dcdsb.2020136.  Google Scholar

[19]

G. Floridia, C. Nitsch and C. Trombetti, Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states, ESAIM Control Optim. Calc. Var., 26 (2020), 34 pp. doi: 10.1051/cocv/2019066.  Google Scholar

[20]

M. Fotouhi and L. Salimi, Controllability results for a class of one dimensional degenerate/singular parabolic equations, Commun. Pure Appl. Anal., 12 (2013), 1415-1430.  doi: 10.3934/cpaa.2013.12.1415.  Google Scholar

[21]

M. Fotouhi and L. Salimi, Null controllability of degenerate/singular parabolic equations, J. Dyn. Control Syst., 18 (2012), 573-602.  doi: 10.1007/s10883-012-9160-5.  Google Scholar

[22]

G. Fragnelli, Interior degenerate/singular parabolic equations in nondivergence form: Well-posedness and Carleman estimates, J. Differential Equations, 260 (2016), 1314-1371.  doi: 10.1016/j.jde.2015.09.019.  Google Scholar

[23]

G. Fragnelli, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Generators with interior degeneracy on spaces of $L^2$ type, Electron. J. Differential Equations, 2012 (2012), 1-30.   Google Scholar

[24]

G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients, Adv. Nonlinear Anal., 6 (2017), 61-84.  doi: 10.1515/anona-2015-0163.  Google Scholar

[25]

G. Fragnelli and D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, Mem. Amer. Math. Soc., 242 (2016), 84 pp. doi: 10.1090/memo/1146.  Google Scholar

[26]

G. Fragnelli and D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Adv. Nonlinear Anal., 2 (2013), 339-378.  doi: 10.1515/anona-2013-0015.  Google Scholar

[27]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of evolution equations, Lect. Notes Ser., 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar

[28]

M. González-Burgos and L. De Teresa, Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force, Portugal. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.  Google Scholar

[29]

A. Hajjaj, L. Maniar and J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electron. J. Differential Equations, 2016 (2016), 1-25.   Google Scholar

[30]

A.Y. Khapalov, Global non-negative controllability of the semilinear parabolic equation governed by bilinear control, ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 269-283.  doi: 10.1051/cocv:2002011.  Google Scholar

[31]

J. Le Rousseau and G. Lebeau, On carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.  Google Scholar

[32]

J. Salhi, Null controllability for a singular coupled system of degenerate parabolic equations in nondivergence form, Electron. J. Qual. Theory Differ. Equ., 13 (2018), 1-28.  doi: 10.14232/ejqtde.2018.1.31.  Google Scholar

[33]

J. Vancostenoble, Global non-negative approximate controllability of parabolic equations with singular potentials, in Trends in Control Theory and Partial Differential Equations, , Springer INdAM Series, 32, Springer, Cham, 2019,255–276.  Google Scholar

[34]

J. Vancostenoble, Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 761-790.  doi: 10.3934/dcdss.2011.4.761.  Google Scholar

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